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Orbital Exchange Calculations: an Alternative View of the Chemical Bond Paul B

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Orbital Exchange Calculations: an Alternative View of the Chemical Bond Paul B Orbital Exchange Calculations: An Alternative View of the Chemical Bond Paul B. Merrithewa PO Box 120, Amherst, New Hampshire 03031-0120 USA Abstract The purpose of this paper is to explore a model of the chemical bond which does not assume that the electrons of the chemical bonding electron pair can be unambiguously identified with either the left hand or right hand of the bonding atoms when their orbitals overlap to bond. In order to provide maximum flexibility in the selection of the electron’s orbitals, the orbitals have been represented as spatial arrays and the calculations performed numerically. This model of the chemical bond assumes that the identifiability of the bonding electrons is a function of 1-(overlap/(1+overlap)) where the overlap of the two bonding electron’s orbitals is calculated in the usual manner. The kinetic energy of the bonding electron pair and the energy required to meet the orthogonality requirements, mandated by the Pauli principle, are a function of overlap/(1+overlap). The model assumes that the bonding orbitals are straight-forward atomic orbitals or hybrids of these atomic orbitals. The results obtained by applying this simple approach to eleven di-atomics and seven common poly-atomics are quite good. The calculated bond lengths are generally within 0.005Å of the measured values and bond energies to within a few percent. Bond lengths for bonds to H are about 0.02 Å high. Except for H2, bond lengths are determined, independent of bond energy, at that point where overlap/(1+overlap) equals 0.5. I. INTRODUCTION The purpose of the work described here is to test a model for the chemical bond that assumes that the two electrons of the bonding pair are not completely identifiable, with respect to their source atom, when their orbitals overlap. Were the bonding electrons not identifiable in the overlap region it would be impossible to unambiguously identify every element of the orbital’s distribution with either the atom on the right hand or the atom on the left side of the bond. Since the identification is ambiguous, nature would allow one to make the most energy favorable identification, which is to picture the electrons spread out over the whole bond when calculating the kinetic energy. Distributions are representations of order (as differentiated from disorder or randomness). Narrow distributions represent high order, broader distributions less order. When two distributions overlap the order associated with the system decreases with their overlap because, in the overlap region, one can no longer identify each distribution element with a particular distribution. Overlap introduces randomness. Electron kinetic energy relates to order and electron distributions to representations of order. 1 In his early work applying LCAO-MO theory to H2, Coulson assumed that the two electrons of the bonding pair were identifiable. In subsequent applications of LCAO-MO theory to this molecule and more complex molecules, the same assumption has been made2. This assumption apparently has its origin in the view that a single electron can be uniquely assigned to one of two non-orthogonal overlapping spatial orbitals of different spin. a Electronic mail: [email protected] 1 That the electrons of two overlapping orthogonal spatial orbitals of the same spin are not distinguishable is manifest in the so-called “exchange integral”3. The exchange integral evaluates the electron-electron repulsion in the overlap region. The total electron-electron repulsion is reduced by this quantity (Since an electron cannot repel itself.). Likewise, it is reasonable to hypothesize that the electrons of the bonding pair, which have non-orthogonal spatial orbitals of different spin, are also not distinguishable in the overlap region. A. Numerical Methods The calculations described herein are done numerically with the orbitals represented in huge identical spatial arrays of the form phi[i][j][k] where i is the index for the radius from the molecular axis, j represents the distance along the molecular axis and k is an array identifier. (Only two spatial coordinates are needed since the electron structure is axially symmetric or can be treated as such.) In a numerical calculation, integrals become summations. For example, the electron-electron repulsion becomes: electron-electron repulsion = constant 퐼푀퐴푋 퐽푀퐴푋 퐼푀퐴푋 퐽푀퐴푋 ∑푖=0 ∑푗=0 ∑푙=0 ∑푚=0 phi[i][j][left array] phi[l][m][right array] ovr[ndist][i][l] , (1) where ovr[ndist][i][l] is the reciprocal of distance between phi[i][j][left array] and phi[l][m]{right array] and ndist = absolute value(-j+m). The value of the constant depends on the dimensions associated with the array elements. As a practical matter, for the repulsion calculation, adjacent phi[i][j] and phi[l][m] are consolidated to reduce the number of array elements. The electron-electron repulsion calculation requires less resolution than the kinetic energy calculation described below. Numerical methods are advantageous because they do not presume a particular functional form for the orbitals. These numerical methods are particularly advantageous in making bonding orbitals which are orthogonal to the opposite core 1s electrons. Numerical methods permit the use of iterative analytical methods to find the lowest energy orbitals which meet the orthogonalization requirements. B. Kinetic Energy of Combined Orbital Kinetic energy (KE) of an orbital is determined in the usual manner: KE = ∫∫2 2 r dr dl, (2) where represents a generic orbital and 2 is the Laplacian. The variable r represents the radial distance from the bond axis and l represents the distance along the bond axis. Using numerical methods this becomes: 퐼푀퐴푋 퐽푀퐴푋 KE =constant ∑푖=0 ∑푗=1 ((rrdn[i] (phi[i − 1][j] − phi[i][j]) − rrup[i](phi[i][j] − phi[i + 1][j]))) phi[i][j] + 퐼푀퐴푋 퐽푀퐴푋 constant ∑푖=0 ∑푗=1 ((phi[i][j + 1] − phi[i][j]) − (phi[i][j] − phi[i][j − 1])) (phi[i][j] rr[i]) , (3) where rr[i] is the axial radius at i 2 rrdn[i] is one-half unit down the axial radius from rr[i] rrup[i] is one-half unit up the axial radius from rr[i]. (In the equation above i=0 leads to phi with a negative index (and therefore not determined). This is of no consequence because its factor, rrdn[i] = 0 when i=0.) The value of the constant depends on the dimensions associated with the array elements. As a practical matter, to increase accuracy while not sacrificing processing speed, finer arrays have been created for areas close to the nucleus (where phi changes faster) and coarser arrays at a distance from the nucleus. Another important quantity is the overlap of the bonding orbitals which is calculated in the usual manner: overlap = ∫∫l r 2 r dr dl, (4) where l represents the bonding atomic orbital on the left and r the bonding atomic orbital on the right. The orbitals l and r are atomic orbitals which have been made orthogonal to the core electrons on the opposite atom. Overlap has a range from 0.0 to 1.0. The atomic orbitals which have been made orthogonal to the core electrons on the opposite atom have no net overlap with those core-electron orbitals: ∫∫l 1sr 2 r dr dl =0.0 and (5) ∫∫r 1sl 2 r dr dl =0.0. (6) To calculate the kinetic energy when the identification of components of the orbitals as right versus left is ambiguous, one creates a hypothetical orbital, referred to as the combined orbital, l+r which distributes the electron equally on the atoms left and right of the bond while preserving the electron density map of the original left and right atomic orbitals combined and meeting the orthogonality requirements. For each spatial array element one computes: phi[i][j][combined] = square root (phi[i][j][right_ortho] phi[i][j][right_ortho]+phi[i][j][left_ortho] phi[i][j][left_ortho]) . (7) When phi[i][j][left_ortho] and phi[i][j][right_ortho] are negative, a negative sign is given to the combined orbital. The array elements phi[i][j][left_ortho] and phi[i][j][right_ortho] are the array elements of atomic orbitals which have been altered to create orbitals which are orthogonal to the core electrons of the opposite atom (l and r). The analytic procedure used to convert the set of array elements, phi[i][j][left] and phi[i][j][right], the original atomic orbitals elements, designated as l and r, to the set phi[i][j][left_ortho] and phi[i][j][right_ortho] is described in Section IIIA below. Note that the combined orbital is not the sum of the right and left orbitals as this would change the electron density map by adding electron density between the atoms. The author knows of no explicit functions which describe the set phi[i][j][left_ortho] and phi[i][j][right_ortho] or l+r. These calculations can only be performed numerically. The kinetic energy change associated with overlap of the atomic orbitals is designated KEbond. KEbond is calculated: KEbond = (overlap/(1+overlap)) KEnet (8) where KEnet=(KEl+r - KEl - KEr) (9) 3 The total kinetic energy change for the bond is 2.0 times KEbond. (KEbond is for one electron.) This approach to the kinetic energy assumes that, to the extent of overlap/(1+overlap), the electrons are spread over the bonding orbitals of both of the bonding atoms. To the extent of 1- (overlap/(1+overlap)), the kinetic energy is that of the atomic orbitals which have been made orthogonal to the opposite core electrons (the set phi[i][j][left_ortho] and phi[i][j][right_ortho]). C. H2 Applying this approach to find the kinetic energy of the H2 bonding electrons, I find a bond energy, De, of 4.06 eV at a bond length of 0.753Å using an orbital scale factor (orbital reduction factor) of 1.15.
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